Hackman Spheroidal Tmatrix2 dsdssd

An application of the spheroidal-coordinate-based transition matrix: The acoustic scattering from high aspect ratio solids

Roger H. Hackman and Douglas G. Todoroff Naval Coastal Systems Center, Panama City, Florida 32407

(Received 29 March 1984; accepted for publication 22 April 1984}

In a previous paper [Roger H. Hackman, J. Aeonst. Soe. Am. 75, 35-45 (1984}], a spheroidal- coordinate-based transition matrix formalism was established for acoustic and elastic wave scattering. In this paper, the acoustic scattering by a solid elastic cylinder with hemispherical endcaps and a length-to-diameter ratio of 10 is considered. Numerical results are presented for the backscattered form function as a function of frequency for various angles of incidence. These results are compared with experimental measurements taken at the Naval Coastal Systems Center and given a physical interpretation.

PACS numbers: 43.20.Fn, 43.30.Gv

INTRODUCTION

In 1968, Waterman' introduced a new technique for systematically solving the acoustic diffraction problem for an arbitrarily shaped body, the transition matrix approach. For the sake of completenesss, we briefly outhne the development of this formalism below. We assume that the acoustic field satisfies the standard Helmholtz equation

•'(r') -I- f, ds h.[•+VG (k I r -- r'l) -- G(k I r -- =

10, r'•v,

where •3 is the total acoustic field, •b i is the incident field, G is the free space Green's function and the integral is over the surface of the scatterer. The subscript ( -I- ) denotes a boundary value as the surface is approached from its exterior and the two different values of the right-hand side of the equation refer to points exterior and interior to the scattering volume, v, respectively. We note that the latter case, i.e., when the field of the reradiating sources induced on the surface of the scatterer exactly cancels the incident field, is often referred to as the "extinction theorem. "• Waterman's procedure is to expand the incident and scattered waves, •3 • and •b' ---- • -- •, and the Ctreen's function G, in regular and outgoing solutions to the scalar Helmholtz equation in spherical coordinates. Since the Green's function is singular when the source and field points coincide, for the extinction theorem the expansion procedure can be implemented only for field points which are interior to the largest sphere which can be inscribed in the scatterer and centered on the origin. However, • is regular throughout the interior of the scatterer. Thus, by the process of analytic continuation, it follows that • must vanish not just in the interior of the inscribing sphere, but throughout the interior of the scatterer. An infinite, coupled set of equations is developed from these results by expanding the unknown surface fields in a suitable basis set. A formal solution to this set of linear algebraic equations can be straightforwardly obtained, giving the expansion coefficients of the outgoing wave in terms of those of the incident field.

This procedure has a distinct advantage over competing approaches in that it appears to have the strongest theoretical foundation? s The approach is formally exact and com- putationally efficient (where applicable), and both the uniqueness 3 and the convergence 4'• of the procedure can be proven for sound-soft and sound-hard scatterers for appropriate choices of basis functions. This is in marked contrast to the usual integral equation approach, 6 which leads to equations which are singular at certain discrete frequencies corresponding to the interior eigenvalue problem, and to those approaches embodying the ad hoc Rayleigh hypothe- sis. ? The T-matrix formalism has been extended to elastic

wave scattering by Waterman s and Pao and Varatharajulu 9 and to acoustic (i.e., scalar wave) scattering from an elastic target by Bostr•m.•ø

It has been subsequently determined that this approach, although formally exact, suffers from a severe practical numerical limitation. For scatterers which deviate strongly from a spherical shape the number of terms required in the expansion of the surface fields increases dramatically and the matrices tend to become ill conditioned. This behavior

had been predicted by Lewin, • who suggested that the source of this difficulty is a hypersensitivity of the surface field to minute errors of the field on the inscribed sphere. That is, assume that the extinction theorem is satisfied to a given order of approximation on the inscribed sphere with a given expansion. The further this spherical surface must be deformed to conform to the surface of the scatterer, the greater the error in the extrapolated value of the field and its derivatives on the surface and hence, the greater the induced error in the outgoing field.

Bates and Wall •2 were the first to suggest that these dif- ficulties could be alleviated by formulating the scattering problem in alternative expansion functions which "better fit" the scatterer. These authors examined the efficacy of a two-dimensional elliptic-cx•ordinate-bascd transition matrix for the scattering of scalar waves from large aspect ratio scatterers. They observed that choosing a coordinate system, such that the "radial coordinate = constant" surface lay as close to the surface of the scatterer as possible, alleviated the

1058 J. Acoust. Soc. Am. 78 (3), September 1985 1058

ill-ctmdititming problems. In later work, Wall l• attempted to develop a transition matrix in prolate spheroidal coordinates for vector wave scatter•g, but was unable to construct a Green's function exposion suitable for general waves. He did succeed in constructing a Green's function for axisymmetric waves, •4 and later applied this formalism to a cylindrical antenna problem. •z While other authors •c'-2• have suggested techniques for improving the numerical stability of the spherical-coord•inzte-based T matrix, none of these approaches holds the promise or the intuitive appeal of the spheroidal-coordinate-based formalism for large aspect ratio scatterers.

The difficulty in constructing a Green's function in spheroidal coordinates lies in the nonseparable nature of the vector Helmholtz equation in this coordinate system. As a consequence of this nonseparability, the vector wave func- titms are not orthogonal and the standard techniques for constructing the Green's function become intractable. WalP 4 succeeds in constructing an orthogonal set of spheroidal vector basis functions by considerin^g grad(•l for the longitudinal degrees of freedom, and curl(•v) for the transverse degrees of freedom, where • and;• are solutions to the scalar Helmholtz equation. However, he succeeds only at the cost of generality---as indicated above, these basis functions can- not describe off-axis scattering processes.

As Waterman has pointed out, 2• Huygen's principle is the funclamcntal concept underlying the T-matrix formalism, and the Green's function technique outlined above is simply one way of obtaining the T-matrix equations. Water- man 23 and Pao • have developed a simple alternative to the above development for displaying the mathematical content of Huygen's principle for vector waves in spherical coordinates. This technique is based on Betti's identity

- = o, where u andv are solutions to the vector Helmholtz equation and where • is the assaciated stress tensor. In an earlier

paper •z (paper I), the covariant generalization of this technique was used to establish a transition matrix formalism for the scattering of general scalar or vector waves in prolate spheroidal coordinates. For vector waves, this approach is far more convenient than that originally adopted by Water- man, especially for coordinate systems in which the vector Helmholtz equation is not separable, as it obviates the need for constructing the vector Grecn's function. Thus the orthogonality of the vector basis states is no longer an important consideration. In the above work, the spheroidal vector basis functions were generated by considering grad(•) for the longitudinal degrees of freedom and curl{r• ) for the transverse degrees of freedom. Here again, •b and yg are solutions to the appropriate scalar Helmholtz equations. We shah refer to this choice of basis functions as the "standard'" choice. 26

Like the standard spherical-coordinate-based approach, this formalism is suitable for scattering problems involving arbitrarily shaped bodies. Now however, we have the capability to explicifiy tailor the formalism to the aspect ratio requirements of the scatterer. In the present paper, we consider a specific application of this approach to acoustic scattering from a finite, solid elastic cylinder to determine its efficiency in dealing with slender, elongated scatterers.

There have been a number of previous formulations of acoustic scattering problems in spheroidal coordinates. Ko- tani, •? who is perhaps the earliest example, considered the acoustic diffraction by a circular disk in oblate spheroidal coordinates. Bowkamp, •s and later Spence, •9 gave more thorough, rigorous treatments of this problem and of the diffraction by a circular aperture. Spence and Granger • are generally accredited with formulating the scattering from a rigid prolate spheroid as a modal harmonic sum in spheroidal coordinates. These authors presented numerical results in the form of beam patterns for various aspect ratios and angles of incidence for low frequencies. Senior •! gave numerical results for rigid and pressure release spheroids for axially incident acoustic waves, and Kl'•schchev and Sheiba • extended these results to oblique incidence. There hav• been a number of solutions for the diffraction of plane waves in the high frequency regime? -z7 The more predomi- nant approach z•-z• has been to use an extension of the Wat- son transformation to convert the modal harmonic series for

sound-hard and sound-soft spheroidal scatterers into a resi- due series which converges more rapidly at high frequencies. Lauchle, z? however, directly evaluated the harmonic series by introducing high-frequency asymptotic expansions for the spheroidal functions.

Silbiger •s was the first to consider the acoustic scattering by a penetrable spheroid in spheroidal coordinates. His treatment was incomplete in that the result was formulated in terms of a surface impedance operator Z, which is in general unknown. While Z was evaluated for several special cases, no attempt was made to incorporate the full complex- ity of an elastic prolate spheroidal scatterer. Yeh 3ø considered the scattering from penetrable liquid (i.e., no shear degrees of freedom) prolate and oblate spheroids and gave numerical results for the special case that mode coupling is not present, i.e., when the sound speed of the scatterer is identical to that of water.

All of the above calculations involve only sc•_ !or fields and the boundary conditions can be implemented in a natu- ral fashion due to the orthogonality of the solutions to the scalar Helmholtz equation in spheroidal coordinates. For- marty, the acoustic scattering from elastic targets and the scattering of elastic waves (in elastic media) are far more difficult, as the transverse polarization degrees of freedom require the introduction of the prolate spheroidal vector wavefunctions. As a consequence of the nonorthogonality of the vector wavefunctions, the boundary conditions involving vector waves become relatively intractable in this coordinate system. Although a number of authors have dealt with elastic waves in spheroidal shells and solids, •-• it was not until after 1970 that the acoustic scattering from an elastic body was treated. In a series of papers, Grossman et al., •6 Gutman and Kl'•shcbev, •? and Kl'•shehev •s considered the acoustic scattering from a spheroidal elastic shell. These authors make the same choice for the vector basis set as the

present author, in paper I. To a considerable extent, the history of the formulation

of electromagnetic scattering in spheroidal coordinates par- allels that of the acoustic scattering problem. The earliest attempt was in 1927 by MSglich, •9 who considered the dif-

1059 J. Acoust. Soc. Am., Vol. 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1059

fraction of a plane polarized wave incident on a perfectly ß conducting circulor disk. His treatment was incorrect, however, and the problem was not properly dealt with until 1948 by Meixner. •ø Meixner z• and Flammer s2 later formulated the scattering of an electromagnetic wave by a conducting prolate spheroid in spheroidal coordinates for the special case that the source is a dipole oriented along the axis of the spheroid. Both of the above problems can be reduced to scalar form and the vector character of the electromagnetic wave need not be confronted. The more demanding problem of the scattering of a plane polarized electromagnetic wave by a conducting prolate spheroid, which required the use of the prolate spheroidal vector wavefunctions, was first formulated by Schultz, sz for end-on incidence, and Rauch s4 extended this work to oblate spheroids. Siegel et al? -•6 performed calculations based on Schultz's formalism and presented results for several {low) frequencies. These authors did not make the "standard" choice for the vector basis func-

tions, i.e., that of paper I, but instead chose to represent the vector fields in rectangular coordinates with each of the components given a representation in terms of the solutions to the scalar Heimholtz equation. There was little further activity on this scattering problem until 1975, when Asano and Yamamoto •? formulated the general problem of the scattering of a plane electromagnetic wave with an arbitrary angle of incidence on a dielectric spheroid. These authors' treatment of the problem differs from that of Schultz primar- ily in the use of the standard vector basis functions to represent the E and H fields. Numerical results were presented in the form of angular distributions for both oblate and prolate spheroidal scatterers for several different aspect ratios for low frequencies. In a subsequent series of papers, Asano, zs Kotlarchyk et al., •9 Asano and Sato, •ø and Asano 6• applied the model to an ensemble of randomly oriented scatterers.

This brief review would not be complete without men- tioning the pioneering work of Van Buren et al., • King et al., •s and King and Van Buren •4 on the generation of spheroidal functions, and the later improvements of Patz and Van Buren. • The avai•bility of reliable, efficient codes for the calculation of the spheroidal functions has had a significant impact on the present work.

The first important result of the present work is the de- monstration that the "standard" set of prolate spheroidal, vector basis functions is overcomplete. The (•r,m;l } = (e,O,O), r ---- 1,2 transverse vector basis states can be written as linear combinations of the remaining (•r----e,m-----0,/•0) transverse vector basis states. It follows that all previous calculations utilizing these functions are suspect. This point is dis- cnssed further in Sec. I.

In Sec. II, we discuss the application of our approach to the acoustic scattering from elastic solids. To validate the theoretical procedures and computer codes, we have made a detailed experimental study of the acoustic scattering from a finite, 10:1 aluminum cylinder with hemi.•pherical endcaps (Sec. III). Section IV is devoted to a comparison of the predictions of our approach with the experimental results and to an analysis of the elastic excitations underlying the more prominent features in the backscattered form function.

In Appendix A we give explicit expressions for the stress

tensor in prolate spheroidal coordinates and in Appendix B, we derive the particular variant of the Moore-Penrose in- verse used in the present work.

I. THE vECTOR BASIS FUNCTIONS

The prolate spheroidal, vector basis functions •., used here were defined in paper I. We remind the reader that r = 1,2 refers to transverse degrees of freedom, r ---- 3, to the longitudinal degree of freedom, and that n refers to the set (•,m,l) required to specify the particular solution to the scalar Helmholtz equation. In that paper, it was shown that these functions satisfy the relations

•ds It(Re •)-Re •,•,. -- t(Re •.).Re •0, ] =0, (1) . ds [t(½.,)-•be., -- t(•e., ).½., ] = 0, (2) •ds [ t(•.)-Re ),,.. -- t(Re ½,,.).•.]

= (3)

where 0 •. is a real, symmetric matrix. In eases wbe• the meaning will not be obscured, we shall drop the reference to 7'.

These relations were used in conjunction with Betifs third identity to derive a mathematical representation of Huygen's principle, exterior to

-- i • O..,,a., = • ds [t+-•. -- t(½.ku+ ], 14) iZo...f.. =f. ds [t+-Re ,. -- t(Re ½.).u+]. (Z) and interior to

0 ---- • ds [ t_.Re •o. _ t(Re •)-u _ ], (6)

(7) - '2, b. = - 1, the scattering region. Here, { a. }, { f. }, and { b. } are the expansion coefficients of the incident, scattered, and refract- ed waves, and the superscript 0 denotes quantities pertaining to the scattering region. The -I-(--) subscripts refer to boundary values taken at the exterior (interior) of the boundary of the scatterer, and s denotes that boundary.

The matrix O •, is the cornerstone of our approach and its detailed properties are of some interest. For r = 3, O t• is diagonal in I and independent of m, i.e.,

0 •! .m = [ (• -J- 2•)/kL ] •l,'. (8) For the transverse degrees of freedom, O •.. takes the form

O ],."' = O •.' ---- (p/kr)/2 •'.(hr), (9) where • •, is a parity conserving angular integral which is easily calculable and may be expressed in terms of the expansion coefficients of the angular spheroidal functions. In the spherical limit (f--,O, •--•oo with f•-•kr),

lira •2 ;.(hr) ---- 6n', (10)

1060 J. Acoust. Soc. Am., Vol. 78, No. 3, Septem. ber 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1060

and Eq. (3) is simply an expression of the orthogenality of the vector spherical basis functions. In the present case, where •r is,/n genera/, nonzero and the f•nctions are nonortho- gonal, O t•' is nonetheless intimately involved with the linear independence of the basis functions [ •, }. That is, if

6. =0,

then by Eq. (5), the necessary and sufficient condition for the v_anlshing of the { f, } is that O t•' be invertible.

We have examined the condition of O t•. by performing eigenvalue decomposition of/2 •t, under varying levels of truncation of the expression 66

N

l' --0

with r --= 1,2 and 0<m < 15. For sufficiently small values of N (the exact value depends upon hr) the eigenvalues of/2 are positive and/2 is a well-conditioned matrix. As 3/increases, however, the smallest eigenvalue of/2 •. decreases to zero (to machine accuracy); that is, the m ---- 0 basis functions become linearly dependent. The remaining/2 •,•ø are well conditioned. As N is increased further (we have examined up to N-- 40), the system remains stable in the sense that there is only a single eigenvalue consistent with zero. The remaining eigenvalues are positive and asymptotically approach unity with increasing order at a given level of truncation.

At small values ofhr, the null state ofO • is predomin- antly (•r,m,/) ---- (e,0,0). This perhaps could have been antici- pated since this state vanishes identically in the spherical limit. While the I value of the largest component of the null state generally increases with At, there remains a significant ! ---- 0 component throughout the range of h r to be considered in this paper.

The ideal solution to this problem is to work with the eigenvectors of O i•" and remove the null state from consideration. The technique we adopt in Sec. II accomplishes this. We note, however, that as an alternative, we may simply remove the l ---- 0 transverse degrees of freedom. The result- ing, truncated version of O i•" is well conditioned.

II. THEORETICAL CONSIDERATIONS

We begin by out!ini•g the derivation of the spheroidal- coordinate-based transition matrix for the acoustic scatter-

ing from an elastic solid immersed in an inviscid fluid of infinite extent. We assume the usual boundary conditions

•-u+ = •-u_, (13)

•.t+ = •-t_, (14)

•xt_ =o, apply at the surface of the scatterer. To account for the acoustic penetration of the scatterer, we introduce the expansion of the elastic displacement

(the superscript 0 denotes quantities pertaining to the scatterer) and Eqs. (4) and (5) become

a,, =i• R,,,. bn., 117)

f,, = -i Z R,,, ,. b,,., (18)

with

- xv*. .Re ], (lO) •, = R,, (•, •Re •,). 120)

To ob• a fo•! •lufion • •s. {17) and {18), we must now im•se the • X t = 0 •unda• condition to r•ucc the numar of deg• of f•dom of the eldtic field. It w• •o• by B•m •ø that t•s may • a•mpHsh• by usag- e. (6) m •!a• the eldtic displa•men•, •. (16}, to •e a•ustic n•eld dispigments

u+ = Zc. Re,,. 121) •c r•t is

where

P,,, = • ds [ •.t{Re •o•, ) •-Re ,o•

(22}

+ t(Re •/2)-h X(• XRe ½,o.)], (23)

M,,. = f, ds •.t(Re •n ) ,•-Re •... (24) Combining the above results, we obtain the transition matrix

r = - {• = g P -•M, (26} • = • P -•M. (27) Explicit expre•ions for the displacements and stress

tensors required for the evaluation of P, M, and R in Eqs. (19), (20}, (23), and (24) are given in paper I {displacements) and in Appendix A (stress tensors). We fix the interfocal distance (2f) of our prolate spheroidal coordinate system for a particular scatterer by

f= [(aspect ratio) • -- l]m/{aspect ratio).

With this choice, the spheroid • = l/fexacfiy inscribes the scatterer.

The integrations in Eqs. (19), (20), (23),.and (24) were performed using Gaussian quadrature. The matrix inversion implied in Eq. (25) was implemented by writing

(a superscript T denotes the matrix transpose) and then using •Gauss elimination to solve for T r, after scaling both Q and Q. This procedure tends to be numerically more stable than a direct invcraion of Q followed by matrix multipUcation.

The remaining inversion

P -•M,

in Eqs. (26} and (27) requires somewhat greater care. A de-


tailed examination of P for solid spheroids and finite cylinders with hemispherical endcaps reveals that this matrix tends to be ill conditioned. A typical eigenanalysis of P yields eigenvalues ranging in size from 10 -26 to 10 +2. Holding the physical details of the scattering problem fixed, we find that the exact spectrum of eigenvalues depends upon the number of expansion terms. Invariably, however, after some level of truncation, increasing the number of terms in the expansion further merely tends to increase the number of eigenvalues roughly consistent with zero. The nonzero elgenvalues (and eigenvectors) are stable, to machine accuracy.

Interestingly enough, the stability of the calculation of the backscattered form function is to some extent, independent of the condition of P. For spheroidal bodies, the form function converges quite quickly and is stable under an almost arbitrarily large increase in the number of expansion terms. The finite cylinder displays a greater sensitivity to the condition of P and at high aspect ratios, when the condition of P has deteriorated sufficiently, the calculation becomes unstable. This latter behavior is probably due to the greater reliance of the cylindrical calculation on off-diagonal elements in P, M, and R.

There is a simple rationale for the relative insensitivity of T to the condition of P. Note that those' elastic states

which are weakly coupled to the nearfield can be n,.o more than weakly coupled to the farfield. Thus, R and R must annihilate the same combinations of the expansion coefficients {b, } as P, and it follows that Tis independent of the existence of these states. The difficulty with the finite cylinder is its greater reliance on off-diagonal matrix elements. In general, the far off-diagonal matrix elements are intrinsically less accurate than the diagonal elements and this annihila- tion process is not accurately performed.

Since it is presumably the presence of the uncoupled (or at most, weakly coupled} [b, } which is the source of the limited numerical stability for the finite cylinder, it should prove numerically advantageous to remove these states from our space. Defining A as the operator which projects into the space in which the matrix P is well defined, and using A to consistently obtain the admissible interior elastic states in Eqs. (17) through (24), we write Q in the form

Q = RA (A rPA )-•A rM. (29) In Appendix B we give our procedure for constructing A and implementing the projection process in our calculations. The method is a variant of the Moore-Penrose pseudoinverse technique? One immediate result of this projection procedure is that the T matrix (for the cylinder) becomes much more nearly symmetric. We find that [Tn. -- T•.•] is consistently smaller than 10 -3 or 10 -n times the magnitude of the largest diagonal transition matrix element. There is essen- tially no effect on calculations for spheroidal bodies.

To compare our theoretical predictions with experimental measurements, we introduce the farfield form function

I I ,-• {L/2} I Po ' where L is the length of the target, Po is the amplitude of the incident acoustic plane wave

Po e•"•' = 4•rPo • itSo•l(h, cos 0•o)

X S •t( h, •',•') je•, ( h,• '), (31)

•d p, is the (•gle•d•t) •plitude of •e •t• wave

xs,.

Here, (Bo,•o) and (B• } •e the sphe•c• anD• wMch de•e the dir•tion of the incident •d scattered wav•, r•ctive- ly, and the ex•nsion •fficients of •e •t• wave •ve • expr• in te•s of the T mat• (T is diagonal and m for a•ymmetfic obj•m). ExpUcifiy intr•g •dpo into •. (30) •d ut•zing the •p•tic r• for- mula for he•t •v•

In g•e•, we ch•e • = •o = 0. •s dc•ifion is •ns•- •nt wi• •e g•eral pr•fi• • the field and r• in a fo• •cti• of unity for a •d sphefic• •tte•r of •di• L/2 in the •-fr•uency •it.

We have chos• • t•t our th•retic• t•h•ques a•st a •i• cyHnd• g•met• for two r•ns. F•t, the f• wav• w•ch • pm•gate on a solid, i•te cylinder have •n ext•sively studi• •d •e well •o•. • Sin• a 10:1 cy•der is sufficiently long •at •e •ndifions on •e md•ps should not •ve • appr•ble eff•t on th• wav•, t•s fa• hel• •nsiderably in the an•- ys•. In the pr•mt •, •c •y two •ntfibu•ng wav• (1 } the lower l•tu&n• m•e, w•ch • • char• • • •i•y sy•etfic wave with disp•ent •m•- nen• only in the radi• •d •ial •r•ions, •d (2) the low- •t flexural m•e, which • g•l de• u•n dir•tions. •e dis•ion c• for th• wav• Fig. 1. •e •gher !on•tudi• •d fiexur• m• have cutoff fr•um&• w•ch •e beyond the •ge •side• h•e and •e •ion• m• •not • excit• a•mti•y.

•nd, this g•met• pm•d• a more s•g•t •t of the fo•sm t• would a spheroid or a su•r spheroid. A spheroid is a s• g•met• for which • the sphefi• and spheroi• b•is functions are ve• well suit•. B•- t '• h• no• t•t even s•l deviations from a spheroi- d• g•met• 1• to a sight det•oration • the nu- mefi• •o• of •e s•d•d sphefi•- •a•-b• T mat•. In •e next s•tion we

our •m•l p•ur•.

III. EXPERIMENTAL

Backscattered acoustic waveforms were acquired from a 10:1 solid aluminum cylinder with hemispherical endcaps


4.0

3.0

2.0

1.C

LONGITUDINAL WAVE

FLEXURAL WAVE /

.5 2'.o ka

FIG. 1. The dispersion curves for the lowest longitudinal and flexural modes on an infinite aluminum cylinder of circular cross section.

as a function of aspect angle O (the angle the acoustic axis makes with the long symmetry axis of the target). The experimental farfield form function was determined using the expression 69

(• ) 2r g,(kL/2,t•) f. ,e =œ72 where r •s the distance from the center of the scattering target to the measurement point, g, (kœ/2,8 ) is the transform of the backscattered waveform, ps(t ), and g•(kL/2) is the transform of the incident pulse at the scattering target, p,(t ).

Acoustic measurements were performed in a 10-X 10- X 7-ft reinforced concrete test pool filled with a suitable volume of freshwater as the acoustic medium. The source transducer, receive hydrophone, and scattering target were placed along a horizontal line, held by a system of precision positioning devices. Lateral, longitudinal, and vertical alignment of the components was accomplished optically using a sighting telescope. The respective positions were such as to maximize the shortest reverberant path, thus optimizing the time domain measurement window. A 6-in.-long target was selected based on the requirements that a sufficient backscattered return would be observed end on, and that a sufficient number of returns from surface waves traveling around the target would be acquired to yield a representative measure of the form function. This latter criteria was not fulfilled in certain cases, as will be discussed at the end of this section. The initial angular orientation (90' off axis) of the scattering target was determined acoustically by maximizing the backscattered signal. Throughout the measurements, the positions of the transducer and hydrophone remained fixed, while the aspect angle of the target was varied.

The relative pressure amplitudes po(t) (of the incident waveform), and p,(t) were measured using a common hydrophone (B & K model 8203}, placed in the farfield region of the transducer, a distance r from the scattering target. Appropriate corrections for spreading and absorption were made to yield an effective pressure wave front, p•(t ), incident on the target. No attempt was made to correct for the slight distortion (estimated to be less than 1%) by the pres-

ence of a receiver hydrophone situated on a straight line between the source transducer and the scattering target.

The drive and receive electronics are depicted in Fig. 2. A single cycle sine wave, amplified by a power amplifier (Kronhite model DCA-50), provided the drive signal for the piezoelectric transducer (USRD type F-33). The acoustic signals received by the hydrophone were preamplified and bandpass filtered prior to digitizing by the HP-5180A waveform recorder.

Data acquisition was under the control of a HP-9826 microcomputer. Digitized representation of the analog waveforms was transferred to the microcomputer for off-line analysis. The data analysis routine allowed for signal averag- ing, convolution, and Fourier transformation of the digitized waveforms, along with graphic displays of the resultant operations. All recorded waveforms were repetitively aver- aged for 30 scans.

Waveforms were sampled and digitized such that the Nyquist rate was always exceeded. An 8192-point discrete fast Fourier transform was implemented with a rectangular window (no weighting) to obtain intermediate frequency domain data. In order to span the kL/2 region of interest (5- 20), several different center frequencies were required. The 3-dB down points were chosen as the cutoff points for the contribution of each frequency band to the computed form function. In this context, we note that many of the form function plots represent averages of numerous independent measurements.

In viewing the individual backscattered waveforms for analysis, it became apparent that some of the returns persist- ed well beyond the available time window. The effect was particularly significant in regions where the theory predicted high Q resonant features. These observations were not surprising in that any experimental measurement has an in- herent limitation on the resolution of such features. Assum- ing a decay modulus (time required for the amplitude to de- crease to 1/e of its initial value) equal to the available

X-Y PLOTTER GENERATOR

I

FREQUENCY GENERATOR I

POWER AMPLIFIER

-• 24'

TRANSDUCER

HP-g826 MICROCOMPUTER

HP-5180A WAVEFORM RECORDER

I

BAND.PASS FILTER

i PRE- AMPLIFIER

I

HYOROPHONE

TARGET

FIG. 2. A schematic representation of the experimental geometry and appa- ratus. Here, 8 defines the experimental aspect angle.

1063 J. Acoust. Sec. Am., Vol. 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1063

window size (minus the specular return}, we estimate that these measurements allow for the correct representation of features with a maximum Q value of five. The particular theoretical resonant features of interest clearly exceed this value.

In order to determine the effect such artificial termina-

tion had on the resultant form function, a simple subtraction was employed for the 30 ø off-axis measurement. Surface multipaths were removed by subtracting the multipath waveform (without target) from the backscattered target return. Although a residual signal was clearly present (giving rise to an interference pattern) and some of the signal still resided outside of the "expanded" measurement window, the agreement with theory was excellent, as will be discussed in the following section.

IV. DISCUSSION

A series of calculations of the monostatic backscattered

form function was performed in the region kL/2 < 20 for comparison with experiment. There are two parameters to fix in the calculation of the m = 0 T matrix at a given frequency, the number of terms included in the truncation of the infinite dimensional matrix equations, Lm• in Eqs. (25)- (27), and the number of significant figures retained in the orthogonalization of P, i.e., No• (see Appendix B). In the frequency range kL/2<16, the number of expansion terms was fixed by increasing L•,• until the form function varied by 0.5% or less for a change AL,• = 2. The number of terms required for convergence with this criterion varied between ten at the low-frequency end and 30 at the upper frequencies. For kL/2> 16, L• was fixed at 30. We note

that for kL/2• 20, a changeAL•,,, = 2 resulted in a change in the form function of less than 2%.

The calculation was relatively insensitive to the number of significant figures retained in the orthogonalization process, provided No, was sufficiently large, and we chose N•h ---- 18--20. Increasing/V• by two generally resulted in a change in the form function of much less than 0.5%.

No attempt was made to measure the density and sound speeds of the aluminum cylinder. Instead, the generic values p -- 2700 kg/m 3, C• = 6420 m/s, and C, = 3040 m/s were used. Thus some chscrepancy between the measured and calculated form function is expected due to the mismatch in the elastic parameters. To obtain some idea of the sensitivity of the calculation to reasonable variations in these parameters, we have examined the position of the lowest resonance for end-on incidence under a 10% variation in the Lame'

parameter/t (the bar speed depends strongly on this parameter). TM We find that the peak shifts by no more than 5%.

In Fig. 3 we exhibit a collection of calculated monostatic beam patterns which have been selected to display the more prominent features/n the backscattered form function. In particular, the single, strong, high/2 lobe at d = 56 ø and kL / 2 = 5.4 and the "ridge" of lobes with d•30 ø beginning at kL/2 = 6.6 represent significant departures from rigid behavior. It is worth noting that with two exceptions, at • / 2 = 5.4 and 10.8, there is no ancillary structure (in the beam pattern} associated with these off-axis lobes. The two exceptions display on-axis resonances. We also note for future reference that the peak amplitude of the ridge, which has large, high {2 peaks at kœ/2 = 6.7, 9.3, and 12, shows a general tendency to move in the direction of increasing 8 with increasing frequency. These features are particularly revealing

.9

Z I

*• 9.3

• • 9.0

•.3 u. 8.6

6.7

5.4

5.0 .3 .6

FORM FUNCTION

.9

9.8

12.8

12.4

12.0

11.6

11.2

10.8

10.4

102

FIG. 3. The calculated monostatic beam

patterns for a 10:1 finite aluminum cylinder with hemispherical endcaps.


of the elastic response of the scatterer and we shall explore this in detail below.

In Figs. 4 and 5, the calculated backscattered form function (solid points) is compared with the experimental data (solid lines) for aspect angles 0 ranging from 0' to 90', in 10' increments. In regions where the experimental frequency bands overlap, both sets of data are presented. The dashed lines in Fig. 5 for 0 = 30', represent measurements taken with the expanded window discussed in the previous section. We note that the small scale oscillations in the data taken

with the expanded window are an artifact of the subtraction procedure and do not have any underlying physical sign/i- cance. In this figure, it is clear that the longer window gives a considerable improvement in the agreement between the data and the calculations. It is also clear that the very high Q features in the calculation (e.g., at kL/2 = 6.7 and 0 = 30') are still beyond the capabilities of the expanded window.

Consider first Fig. 4, i.e., end-on incidence. Since T-matrix calculations for end-on incidence require only the m = 0 term in Eq. {33) and thus tend to be less time cons-ming than off-axis orientations, a somewhat more ambitious calculational program was undertaken here than for more general angles of orientation; hence, the expanded frequency scale and greater density of points, as compared to Fig. 5. The signal strength from the target at this angle was too small to reliably subtract the multipaths so no attempt was made to obtain the high Q features present in the data. Given this qualification, the calculation is in reasonably good agreement with the data. The three peaks at kL/2 -- 5.4, 10.8, and 16.2 are clearly the,• = L/2, L, and 3L/2 resonances associated with the largely nondispersive, lowest longitudinal mode of an infinite elastic cylinder. We have examined the elastic displacements at the surface of the scatterer, and both the calculated phase speed and the displacements are in ac- cord with this picture.

Consider next the prominent ridge of lobes in the backscattered form function depicted in Fig. 3 and in the 0 = 30' portion of Fig. 5. This ridge is associated with the excitation of the lowest flexural wave of an infinite cylinder due to the matching of the trace velocity of the incident acoustic wave with the velocity of propagation of the flexural wave (see Fig. 2). An examination of the elastic displacements reveals that the three clearly identifiable resonance peaks at 0 = 30. in Fig. 5 are associatedwiththeA ---- 2L/5, L/$,and 2L/7 resonant wavelengths. The agreement of the calculation with the data, particularly that data obtained with the expan.ded window, is quite good. However, the very high Q portion of the

resonance peaks is still missing. An examination of the time domain waveform shows that even with the expanded window, only a portion of the energy of these peaks has been captured, i.e., the waveform has been artificially and prema- turely truncated. In the nonresonant region, the flexural wave is still excited, due to the strong coupling mechanism, and the elastic cylinder acts as a phase steered array, firing a highly directional and hence, strong signal back at the source. This mechanism also explains the surprising strength of the resonant response.

The single strong lobe at 56 ø (see the 0 ---- 50. and 0 ---- 60 ø aspect orientations in Fig. 5) is associated with the longitudinal (i.e., m ---- 0) response ofthe cylinder. An examination of the elastic displacements reveals a stationary longitudinal wave with,• --- L/2 almost identical to that excited at end-on

incidence at this frequency, but approximately 50% stron- ger. At this angle of incidence l0 --- 56 ø) and frequency, there is exactly one water wavelength along the projected length of the cylinder and it is the coherent shaking of the two ends which is responsible for the relatively strong excitation. Un- fortunately, due to the very high Q nature of this lobe and the small signal strength, even with the expanded window only a small enhancement ofthe form function was observed at this

angle. Therefore the data and calculations were not included in the figure.

Consider finally, the broadside form function. The frequency region considered here is too low for any significant elastic activity in the broadside or near broadside direction (70ø,•90ø). It is interesting to note that the off-axis behavior in this angular region differs markedly from that obtained in studies of the scattering of obliquely incident acoustic waves by an infinite cylinder by Flax et al. ?• How- ever, the lobe structure in Fig. 3 in this angular region can be simply explained'by treating a finite rigid cylinder of length L as a linear phase steered array excited by the incident acoustic wave. The primary, secondary, and tertiary lobes are all present and their position, angular width and relative magnitude are consistent with this simple picture, although the third lobe is somewhat larger than expected. We note that there is sufficient experimental sensitivity to the angular alignment in this region, due to the relatively narrow (in 0 ) lobes, to account for the small discrepancies between the data and calculations.

We note in closing'that, in related work, Suet al.?2 and Numrich et al.?3 have compared with experiment the predictions of the standard, spherical-coordinate-based T matrix for a finite aluminum cylinder of small aspect ratio (2:1).

_o i-

• .1,3

2 4 6 8 10 12 14 16 18'

FIG. 4. The backscattered farfield form

functions for a 10: ! finite aluminum cylinder with hemispherical endcaps for end-on incidence, plotted as a function of the dimensionless frequency/eL/2. The heavy dot• repr•ent calculat•l vnl= ues and the solid lines, experimental value• Note that in some regions, several frequency bands overlap.

1065 J. Acoust. Soc. Am., VoL 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1065

z

z

z

2.0

10 • OFF AXiS

1.0

.o

2.0--

1.0

30 ø OFF AXIS

ß :.! 7 9 11 13 15

ß KLJ2

.9

.0

.9

•o 3

OFF AXIS

OFF AXI,•

7 9 11 13 15 KL/2

20' OFF AXIS

40 ø OFF AXIS

7 9 11 13 15 KL/2

OFF AXIS

OFF AXIS

7 9 11 13 15 KL/2

FIG. 5. The ba=kscattcred faff•ld form function for a 10:1 finke alumi-

num cylinder with hemispherical endcaps for off-axis ;n•. The notation is th= same as in Fi& 4. The additional dashed fin•s in the

0- 30' graph represent experlink-

.9 90 • OFF AXIS ß . ß

7 9 11 13 15 KL•

ACKNOWLEDGMEHTS

The authors are indebted to D. H. Trivett for stimulat-

ing and productive conversations throughout the course of this work. One of us (RHH) would also like to thank A. L. Van Buren for discussions pertaining to the generation of the prolate spheroidal functions and for copies of his spheroidal function codes. In particular, the elegant eigenvalue routine 0fPatz and Van Buren • served as the heart of our eigenfunc- tion routine. RHH would also like to thank I. Dobeck for

discussions on the Moore-Penrose pseudoinverse and T. Rackets for producing the dispersion curves in the text. Fin- ally, the authors would like to thank L. Flax for his contin- ued encouragement and support.

APPENDIX A

In applications of the spheroidal-coordinate-based transition matrix, it is necessary to have explicit expressions for the vector basis functions and their associated stress tensors.

Expressions for the basis functions were given in paper I; we give the stress tensors in this Appendix.

Following Morse and Feshbach, TM we intr6duce the independent variables

= ',V,cos ).

The associated unit vectors

1066 J. Acoust. Sec. Am., VoL 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scatter'ng from ratio solids 1066

define a right-handed coordinate system. This differs from the definitions adopted in paper I which utilized a left-handed coordinate system. Thus, there are several minus sign differences between the vector basis functions given in paper I and those adopted here, and these differences are noted in Eq. (A4). For calculational purposes, it is convenient to work with a representation of the vector basis functions in which the "physical" components are utilized, i.e.,

3

• __- • (•.),•,. (A3) The relation between these physical components and the co- vnziant expressions given in paper ! [the (•.)• ], is

(•), = + (h,)-'(•),,

where the upper sign is for ½ -- 1, 3 and the lower, for •- -- 2. The h• denote the scale factors

h, =f[(• 2 _ •/2)/(• 2 _

- (AS)

ha = (f/sin •b )[(1 - •/2)(• 2 _ 1)1,/2. The stress tensor is defined by

•(½,•) =ZV4•i + •(V,• + •,•V), where

is the unit dyadic and where

= 2 • I•x• •,-•/] + •b•.•(ln

x(a,a• + 6a,). {A7)

•t Z• = R•z(fir,• •t(hr,•,• ) denote a pro•rly no•- • •lution to the scal• He•holtz equation with k = kT, •d let •t = •z(hT) denote the eigenvMue. For the trans- vene degr• of freedom, the physical •m•nents of the str•s tensor (•u) •e $ven by

1067 J. Acoust. Sec. Am., Vol. 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1067

In the above expressions, we have eliminated all reference to o•x/v•õ :, c9:X/o•q:, and c9 :•y/•2 through the use of the defining differential equations for R.,• and

Using •. = R,,,t(h•.,g}S•t(ht.,•I,q•) to denote a properly normali7ed solution to the scalar Helmholtz equation with k = k,., and A•,t =.4,,,t(hL) tO denote the eigenvalue, for the irmtational degree of freedom, we find

(AIS)

(AI9)

1068 J. Acoust. So(=. Am., VoL 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1068

=

=

f• [(1-- •7:)• -- •7:)] m • •'e-- 1 &b • '

(A20)

(A21)

(A22)

APPENDIX B

Consider the ill-conditioned problem

PC=M,

where C and M are n X m rectangular matrices and where P is an n X n real, symmetric matrix with one or more zero eigenvalues. Here, C is to be determined. Let Z k be the k th eigenvalue of P and let Pi(k ) be the corresponding normal- ized eigenvector, i.e.,

• Pi• p•lk ) = Ak p,(k ). (B2)

Then the transformation matrix which diagonalizes P is

We assume that P has Nnonzero eigenvalues, and that these cigenvalues are ordered by decreasing magnitude. The in- consistencies may be removed from Eq. (B1) by eliminating the null space of P. This is most straightforwardly accomplished by performing an eigenvalue decomposition of P and projecting into the space spanned by eigenvectors having nonzero eigenvalues; that is, we wish to cast Eq. (B 1) into the form

aN c = M,

where

/2N = ,li 0 0 2N

(BS)

[A,I>O, n= 1,...• r,

and where UN is the truncated diagonalization transformation matrix. However, the explicit construction of the eigenvalues and eigenvectors tends to be time consuming, and we adopt a somewhat different procedure here.

From the spectral decomposition

P,2 = i A•p,(k)p2(k), (B6) k=l

it is clear that ifP is of rank N, then P has exactly N linearly independent columns. To form a basis for this N-dimensional space, we apply the modified Graham-Schmidt orthogonalization procedure to the columns of P, with pivoting. That is, at each step in the orthogonalization procedure, from among the non-normali7ed columns of P, we choose the col- umn with the largest norm, normalize it and orthogonalize

all of the remaining columns relative to this choice. Clearly, this procedure can be applied only N times before we have exhausted the possibilities. In practice, we orthogonalize all columns, then order the columns by their loss of significant digits (through the subtraction process) and truncate at the point where more than 18 significant figures have been lost.

ThecolumnsofthenXNmatrixA thus formed, must be a linear combination of the eigenvectors having I/tk [ > 0, i.e.,

N

Aq = •__ • p•(k ) A• (B7) which in matrix notation is

A = UN .4.

From the orthonormality of the columns of A it follows that the N XN matrix A is orthogonal. From F_x 1. (BS), we thus have

Returning to Eq. (B4) and ut'dizing

A rPA =.4 r.(2 N ̀4 (B10)• we find

{A •PA )A T½=A •M, (Bll) i.e., that A performs the necessary projection process. Note that it is not necessary to construct ̀4.

The exact relation between the loss of significant digits in the columns of P during the orthogonalization process and the s•.• of the magnitudes of the eigenvalucs in •2 N is unclear. This is unimportant, however, since our real con- cern is the linear independence of the columns (and rows) of the truncated matrix.

•P. C. Waterman, "New Formulation of Acoustic Scattering," 1. Acoust. Soc. Am. 45, 1471-1429 (1969).

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•P. A. Martin, "On the Null-Field Equations for the Exterior Problems of Acoustics," Q. J. Mech. Appl. Math. XXXHI, 385-396 (1980); P. A. Mar- tin, "Acoustic Scattering and Radiation Problems and the Null-Field Method," Wave Motion 4, 391-408 (1982).

4A. G. Ram, "Convergence of the T-Matrix Approach to Scattering The- ory," J. Math. Phys. 23, 1123-1125 (1982}.

•G. Kristensson, A. O. Ramre, and S. Strom, "Convergence of the T-Ma- t fix Approach in Scattering Theory II," J. Math. Phys. 24, 2619-2631 (1983).

SFor recent work utilizing this technique, see W.Tobocman, "Calculation of Acoustic Wave Scattering by Means of Helmholtz Integral Equation. I," •. Acoust. Soc. Am. 76, 599-607 ( 1984}; W. Tobocman, "Calculation of Acoustic Wave Scattering by Means of the Helmholtz Integral Equation.

1069 d. Acoust. Soc. Am., Vol. 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff: Scattering from ratio solids 1069

II," •. Acoust. Soc. Am. 76, 1549-1554; W. Toboanan, "Extension of the Hclmhoitz Integral Equation Method to Shorter Wavelengths," preprint.

7See, for example, •. C. Balomey and A. Witgin, "Numerical Comparison of the Green's Function and the Waterman and Rayleigh Theories of Scat- tering from a Cylinder with Arbitrary Cross Section," Proc. IEEE 121, 794-804 (1974), and references therein.

ß 8p. C. Waterman, "Matrix Theory of Elastic Wave Scattering," I. Acoust. SOc. Am. 60, 567-579 {1976).

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•øA. Bostr6m, "Scattering of Stationary Acoustic Waves by an Elastic Im- mersed in a Fluid," I. Acoust. SCc. Am. 67, 390-398 (1980).

nL. Lewin, "On the Restricted Validity of Point Matching Techniques," IEEE Trans. Microwave Theory Tech. MTT-lg, 1041-1047 {1970).

•2R. H. T. Bates and D. L N. WaR, "Null Field Approach to Scalar Diffrac- tion I. General Method," Philos. Trans. R. So•. Lond. Ser. A 287, 45-78 (1977).

•aD. I. N. WaR, Ph.D. thesis, University of Canterbury, England {1976}. •4D. I. N. WaR, "Circularly Symmetric Green Tensors for the Harmonic

Wave Equation in Spheroidal Coordinate Systems," I. Phys. A: Gem 11, 749-757 {1978).

•sD. I. N. WaR, "The Null Field Approach to the Antenna Boundary Value Problem," IEEE International Conf. on Antennas and Propagation. Part I, 174 (1979).

•6D. I. N. WaR, "Methods of Overcoming Numerical Instabilities Associat- ed with the T-Matrix Method," in Acoustic, Electromagnetic and Elastic Wave Scattering, edited by V. V. Vatarian and V. K. Varadan (Pergamon, New York, 1980), pp. 269-286.

"W. M. Visscher, "Method of Optimal Truncation: A New T-Matrix Ap- proach to Elastic Wave Scattering," in Acoustic, Electromagnetic and Elastic Wave Scattering, edited by V. V. Varadan and V. K. Varadan {Per- gatnon, New York, 1980}, pp. 287-301.

'SM. F. Werby and L. H. Green, "An Extended Unitary Approach for Acoustical Scattering from Elastic Shells Immersed in a Fluid," I. Acoust. SOc. Am. 74, 625-630 (1983).

•0A. Bostr6m, "Scattering of Acoustic Waves by a Layered Elastic Obstacle in a Fluid--An Improved Null Field Approach," I. Aeoust. SOc. Am. 76, 588-593 { 1984).

•øA. Lakhtakia, V. K. Varadan, and V. V. Varadan, "Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios," I. Acoust. Soc. Am. 76, 906-912 (1984).

•'P. C. Waterman, "Surface Fields in Potential Theory and Acoustic Scat- tering," ]. Acoust. Soc. Am. 76, 1215-1226 {1984).

•P. C. Waterman, "Survey of T-Matrix Methods and Surface Field Repre- sentations," in Acoustic, Electromagnetic and Elastic Wave Scattering, edited by V. V. Varadan and V. K. Varadan (Pergamon, New York, 19801, pp. 61-78.

23p. C. Waterman, "Matrix Theory of Elastic Wave Scattering. II. A New Conservation Law," I. Aconst. SOc. Am. 63, 1320-1325 (1978).

•4y. H. Pan, "Betfi's Identity and the Transition Matrix for Elastic Waves," I. Acoust. Soc. Am. 64, 302-310 {1978}.

•SR. H. Hackman, "The Transition Matrix for Acoustic and Elastic Wave Scattering in Prolate Spheroidal Coordinates," I. Acoust. SOC. Am. 75, 34-45 {1984}.

• for example, C. Flaremet, Spheroidal WaveFunctions (Stanford U.P., Stanfi•rd, CA, 1957} in this regard.

:?M. Kotani, Proc. Phy•-Math. SOc. Jpn. 15, 30 {1933}. •sC. I. Bowkamp, "Theoretische en Numerieke Bahandeiing van de Bnig-

ins Door Een Vonde Opening," Diss. Groningen, Groningen-Batavia, 1941; also J. Math. Phys. 25, 79 {1947}.

•1•.. D. Spence, "The Diffraction of Sound by Circular Disks and Apera- tures," I. Acoust. Soc. Am. 20, 380-386 {1948).

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•T. B. A. Senior, "The Scattering from Acoustically Hard and Soft Prolate Spheroids for Axial Incidence," Can. •. Phys. 44, 655-667 0966).

•ZA. A. Kl'/shchev and L. S. Sheiba, "Scattering of a Sound Wave by Ideal Prolate Spheroids," Soy. Phys. Acoust. 16, 219-222 {1970).

ZaA. A. Kli/shchev, "Sound Scattering by Ideal Spheroids in the High Fre- quency Limit," SOv. Phys. Aconst. 19, 447-449 {1974}.

•*N. D. K•rinotf and R. K. Pitt, "On the Theory of Diffraction and Its Application to the Prolate Spheroid," Ann. Phys. 6, 227-299 (1959).

Z•R. F. Goodrich and N. D. Kazarinoff, "Scalar D/ffraction by Prelate Spheroids Whose Eccentricities Are Almost One," Proc. Cambridge philos. Sec. S9, 167-183 (1963}.

•B. D. Sleeman, "Integral Representations Associated with High Frequen- cy Non-Symmetric Scattering by Prelate Spheroids," Q. I. Mech. Appl. Math. 22, 405-426 {1969}; B. D. Sleeman, "On Diffraction at Short Wave- lengths by a Prelate Spheroid," J. Inst Math. Its Appl. 5, 02-442 { 1969).

•?G. C. Lauelde, "Short~Wavelength Acoustic Diffraction by Prelate Spheroids," 1. Acoust. Sec. Am. 58, 56g-575 {1975}; /3. C. Lauchle, "Short-Wavelength Acoustic Backscattering by a Prelate Spheroid," J. Acoust. Sec. Am. 58, 576-580 (1975}.

•A. Silbiger, "Scattering of Sound by an Elastic Prelate Spheroki," I. Aconst. SOc. Am. 35, 564-570 {1963}.

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•F. L. DiMaggio and A. Silbiger, "Free Extensional Torsional Vibrations of a Prolate Spheroidal Shell," 1. Acoust. SOc. Am. 33, 56-58 {1961}; A. Silbiger and F. L. DiMaggio, "Extensional Axi-Symmetric Second Class Vibrations of a Prolate Spheroidal Shell," Contract No. Nonr 266{67}, Tech. Rep. No. 3, Columbia University, 1961.

4ZN. Shiraishi and F. L. DiMaggio, "Perturbation Solution for the Axisym- metric Vibrations of Prolate Spheroidal Shells," I. Acoust. SOc. Am. • 1725-1731 (1962}.

42p. I. Nemergut and R. S. Brand, "Axisymmetric Vibrations of Prolate Spheroidal Shells," L Acoost. SOc. Am. $5, 262-265 {1965).

*aF. DiMaggio and R. Rand, "Axisymmetric Vibrations of Prolate Spheroi- dal Shells," 1. iconst. Soc. Am. 40, 179-186 {1966}.

•S. K. Datta and A. H. Shah, "Axially Symmetric Waves in an Elastic Solid of Revolution," J. Acoust. Soc. Am. 44, 473-47 {1968}.

•SR. H. Rand, "Torsional Vibrations of Elastic Prolate Spheroids," I. Acoust. Soc. Am. 44, 749-751 {1968}.

a'O. I. Grossman, A. A. Kl'•hchev, and I. I. Klyukin, "Scattering of an Axisymmetric Plane Sound Wave by a SPheroidal Elastic Shell," Trudy Leningrad, Korabelstr. Inst. ?4, 31-36 {1970}.

•?T. L. Outman and A. A. Kl'/shchev, "Diffraction of Elastic Waves by an Elastic Spheroid," Trudy Leningrad. Korablestroitel. Inst. {Trudy LOLK•) 91, 31-37 {1974}.

4•A. A. Kl'/shchev, ø'Debye Potentials for Three-Dimensional Vibrations of a Spheroid Elastic Shell," Soy. Phys. Acoust. :11, 293-294 {1975}; A. A. Kl'/shchev, "Sound Scattering by an Oblate Spheroidal Elastic Shell," Soy. Phys. Acoust. 21, 571-573 {1976}; A. A. KI//shchev, "sound Scatter- ing by a Spheroidal Body at an Interface," SOy. Phys. Acoust. 25, 78-79 {1979}.

4•C. T. MSglich, Ann. Phys. 83, 609 {1927}. •oj. Meixner, Z. Naturforsch. 3A, 506 {1948}. szl. Meixner, Ann. Phys. 12, 227 {1953). s:C. Flareruer, "Radiation from Electric and Magnetic Dipoles in the Pres-

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$aF. V. Schultz, "Studies in Radar Crom-Section I---Scattering by a Prolate Spheroid," University of Michigan Rep. UMM-42 {19•0}.

S4L. M. Ranch, "Electromagnetic Scattering by an Oblate Spheroid," Uni- versity of Michigan Rep. UMM-116 {1953}.

$SK. M. Siegel, B. H. Gere, I. Marx, and F. B. Sleator, "Studies in Radar Cross-Section Xl--The Numerical Determination of the Radar Cross-

Section of a Prolate Spheroid, University of Michigan Rep. UMM-126 (1953}.

$sK. M. Siegel, F. V. Schultz, B. H. Gere', and F. B. S!eator, "The Theoreti- cal and Numerical Determination of the Radar Cross-Section of a Prolate

Spheroid," IRE Trans. Antennas Propag. AP-4, 266-275 {1956}. a?S. Asano and/3. Yamamoto, "Light Scattering by a Spheroidal Particle,"

Appl. Opt. 14, 29-49 {1975}. asS. Asano, Appl. Opt. lg, 712 {1979}. S9M. Kotlarchyk, S.-H. Chen, and S. Asano, "Accuracy of RGD Approxi-

mation for Computing Light Scattering Properties of Diffusing and Mo- tile Bacteria," Appl. Opt. lg, 2470-24?9 {1979}.

•S. Asano and M. Sato, "Light Scattering by Randomly Oriented Spheroi- dal Particles," Appl. Opt. l•, 962-974 {1980}.

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*:A. L. Van Buren, R. V. Baler, and S. Hanish, "A FOnTRAN Computer Program for Calculating the Oblate Spheroidal Radial IAmctions of the First and Second Kind and Their First Derivatives," NRL Report 6959

1070 J. Acoust. Sec. Am., Vol. 78, No. 3, September 1985 R. H. Hackman and D. G. Todoroff: Scattering from ratio solids 1070

(•an. 1970). 6•B. J. King, R. V. Baler, and S. Hanish, "A FORTRAN Computer Program

for Calculating the Prolate Spheroidal Radial Functions of the First and Second Kind and Their First Derivatives," NRL Report 7012 (March 1970).

taB. $. King and A. L. Van Buren, "A FORTRAN Computer Program for Calculating the Prolate and Oblate Angie Functions of the First Kind and Thdr First and Second Derivatives," NRL Report 7161 (Nov. 1970).

aSgee, for example, B. J. Patz and A. L. Van Burcn, "A FOR TRAN Computer Program for Calculating the Prolate Spheroidal Angular Functions of the First Kind," NRL Memorandum Report 4414 (March 1981).

•"I'here are a number of measures of the condition of a matrix. To be specific in this paper we adopt Von Neumann and C•oldstine's P-condition num-

P•4 ) = •.•l/•,

as the standard for the measure of condition. Here, A• and ,{• are the largeat and smallcat eigenvaluea of the matrix.4 in absolute magnitude. A good gertcanal reference is L R. Westlake, .4 Handbook of Numerical Ma- trix Inversion and Solution of Linear Equations {W'dey, New York, 1968}.

•?See, for example, G. $tran& Linear •41gebra and Its Applications (Aca.

demie, New York. 1976), pp. 96-145. a•The relevant literature is reviewed in $. D. Achenbach, I•'ave Propagat/on

in Elastic •!ids (North-Holland, New York. 1976), pp. 240-249. ø•L. R. Dragonette, "Evaluation of the Relative Importance of Circumfer-

ential or Creeping Waves in the Acoustic Scattering from Rigid and Elas- tic Solid Cylinders and from Cylindrical Shells," NRL Report No. 8216 (197a}.

•øFor example, see Richard K. Cook, "Acoustics," in.4merican Institute of Physics Handbook, edited by D. E. Gravy {McGraw-Hill, New York, 1963}, pp. 3-88. Cook gives a 10% variation in/.• and no variation in•.

?lL. Flax, V. K. Varadan, and V. ¾. Vatarian, "Scattering of an Obliquely Incident Acoustic Wave by an Infinite Cylinder," •I. Aeonst. Soc. 1832-1835 {1980}.

nJ.-H. Su, V. V. Vnradan, V. K. Vatarian, and L. Flax, "Aeoustic Wave Scattering by a Finite Elastic Cylinder in Water," J. Acoust. Soc. Am. 68, 686=691 (1980).

•S. K. Numrich, V. V. Vatarian, and V. K. Vatarian, "Scattering of Aeons- tic Waves by a Finite Elastic Cylinder Immersed in Water," J. Aeonst. Soc. Am. 70, 1407-1411 {1981).

?4p. M. Morse and H. Feshbaeh, Method• of Theoretical Ph3•ics (McGraw- Hill, New York, 1953), pp. 661.

1071 J. Acoust. Sec. Am.. Vol. 78, No. 3. September 1985 R.H. Hackman and D. G. Tedore,f: Scattering from ratio solids 1071

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